NASA Technical Reports Server (NTRS) 20090017701: Acoustic-Structure Interaction in Rocket Engines: Validation Testing

Acoustic-Structure Interaction in Rocket Engines: Validation Testing

R. Benjamin Davis, Scott S. Joji, Russel A. Parks, and Andrew M. Brown
NASA Marshall Space Flight Center, MSFC, AL 35812

Nomenclature

A f Flexible surface area of structure calculated at fluid interface
a Acoustic modal coordinate

c 0 Speed of sound in fluid

E Young’s modulus

[G] Gyroscopic matrix

[K] Stiffness matrix

[L] Acoustic-structure coupling matrix

[M] Mass matrix

q Structural modal coordinate

V Volume of fluid cavity

v Poisson’s ratio

xu Uncoupled natural frequency of acoustic cavity
p 0 Mass density of fluid

p s Mass density of structure

Q In vacuo natural frequency of structure

Subscripts

j Uncoupled acoustic mode number indices

k In vacuo structural mode number indices

Abstract

While analyzing a rocket engine component, it is often necessary to account for any effects that adjacent fluids
(e.g., liquid fuels or oxidizers) might have on the structural dynamics of the component. To better characterize the
fully coupled fluid-structure system responses, an analytical approach that models the system as a coupled
expansion of rigid wall acoustic modes and in vacuo structural modes has been proposed. The present work
seeks to experimentally validate this approach. To experimentally observe well-coupled system modes, the test
article and fluid cavities are designed such that the uncoupled structural frequencies are comparable to the
uncoupled acoustic frequencies. The test measures the natural frequencies, mode shapes, and forced response
of cylindrical test articles in contact with fluid-filled cylindrical and/or annular cavities. The test article is excited
with a stinger and the fluid-loaded response is acquired using a laser-doppler vibrometer. The experimentally
determined fluid-loaded natural frequencies are compared directly to the results of the analytical model. Due to
the geometric configuration of the test article, the analytical model is found to be valid for natural modes with
circumferential wave numbers greater than four. In the case of these modes, the natural frequencies predicted
by the analytical model demonstrate excellent agreement with the experimentally determined natural frequencies.

I. Introduction

Classical acoustic and vibration analysis generally consider acoustic and structural problems independently. This
strategy involves formulating the acoustic problem under the assumption that any adjacent structure is perfectly
rigid. The corresponding structural dynamic problem is then solved by treating the structure as though it is in
vacuo. This approach is generally accurate when the structure is in contact with a light acoustic medium such as
air. However, the interaction between an acoustic fluid and a structure is generally non-negligible when the fluid
is relatively dense and/or the in vacuo structural natural frequencies are comparable to the uncoupled acoustic
natural frequencies. In liquid rocket engines, there exist a variety of flexible structural components in contact with
liquid propellant and/or oxidizer. These liquids have densities many times greater than the density of air.
Furthermore, these liquids often occupy enclosed cavities that possess uncoupled acoustic natural frequencies
comparable to uncoupled natural frequencies of the surrounding structure. When analyzing the dynamics of
rocket engine components, it is therefore often necessary to account for any effects that the fluid might have on
the structural dynamics.

In an effort to characterize the system natural frequencies, modes, and forced response of fluid-loaded rocket
engine components, an analytical approach has been proposed [1], This approach is based on the
acoustoelasticity theory developed by Dowell, et al. [2] which models an acoustic-structure system as coupled
expansion of rigid wall acoustic modes and in vacuo structural modes. Ref. [3] applies this approach to
geometries common to liquid rocket engines. Specifically, Ref. [3] considers cylindrical shells fully coupled to
cylindrical and/or annular cavities filled with acoustic fluid. Figure 1 is a schematic of such as system.

Two Fluid Filled Cavities

Cavity A: Cylindrical
Cavity B: Annular

Figure 1 : Schematic of fluid-structure system under consideration

The present work considers an experimental configuration designed to resemble the system shown in Figure 1.
The experimental results are compared to those calculated by the analytical approach proposed in Ref. [1]. The
experimental results are also compared to results generated from a fully coupled acoustic-structure finite element
model. The following three sections discuss the analytical, finite element, and experimental approaches. The
final two sections of this paper present the results of three different approaches and discuss any relevant
conclusions.

II. Analytical Approach

The equations modeling a flexible structure coupled to an interior enclosed acoustic fluid take the following form

(i)

where

[M]

VMj 0

0 M k

0

A i'Pw \^jk ] 7

jA F C 0

0

w=

VMjtffj

0

The [L jk ] matrices shown here consist of coupling coefficients corresponding to individual pairs of uncoupled
acoustic and structural modes. Each coupling coefficient can be interpreted as a measure of the spatial similarity
between a given modal pair. Eqs. (1) correspond to an unforced, undamped system of unspecified geometry.
Ref. [3] details how Eqs. (1) can be written for the geometry depicted by Figure 1. Ref. [3] further explains how
this system of equations can be solved when damping and forcing terms are applied. In the analytical approach
employed here, the in vacuo frequencies and mode shapes for a clamped-free cylindrical shell are calculated with
a procedure presented by Callahan and Baruh [4]. This procedure efficiently solves the Junger and Feit
cylindrical shell equations [5] for common boundary conditions. The uncoupled acoustic frequencies and modes
for cylindrical and annular fluid-filled cavities are calculated using standard formulae that can be found in (or
derived with the aid of) many standard acoustics text (see e.g., Blackstock [6]).

III. Finite Element Model

Commercial finite element software packages such as ANSYS and NASTRAN have the built-in capabilities to
solve acoustic-structure interaction problems like the one considered here. These capabilities (often called fluid-
structure interaction (FSI) solutions) can be challenging to implement and are often computationally expensive.
An analytical approach, such as the one described in Section II, largely avoids such difficulties. Furthermore, for
cases in which complex geometries necessitate a finite element approach, an analytical solution affords the
analyst the physical insight necessary to interpret and verify the finite element results. While the intent of this
work is to validate the analytical approach presented in [1], an FSI finite element model of the test article was also
created. This model serves as additional validation of the analytical approach. Additionally, after the fabrication
of the test article, it was determined that certain modes are dominated by motion in the base of the test article.
Since the geometric model used with the analytical approach considers only the cylindrical portion of the test
article, these modes are not captured by the analytical calculations. The finite element model thus enables the
calculation of coupled frequencies corresponding to these base-dominated modes.

IV. Experimental Approach

A series of modal tests and sine sweeps were performed on the test article shown in Figure 2. The test article is
a 0.163” thick cylindrical shell with a mean radius of 4.382” and a length of 2.700”. The shell consists of steel (E =
29,000 psi, p s = 0.283 lb/in 3 , v = 0.30) and is welded to a large base also made of steel. A ridge was machined
into the base of the test article in order to accommodate a 0.549” thick acrylic tank. The tank has a mean radius
of 6.060” and a length of 3.704”. An acrylic lid was machined and fastened to the tank with six screws spaced
around the circumference. The design of the experimental set-up allows for the test article to be in contact with
cylindrical and/or annular fluid-filled cavities. Ethyl alcohol (c 0 = 3,960 ft/s, p 0 = 0.0285 lb/in 3 ) was used as the fluid
in all tests. Ethyl alcohol was chosen because it can be handled safely and because it has a density that is
bounded by those of liquid hydrogen and liquid oxygen. To simulate acoustically rigid boundary conditions at the
interface between the lid and the fluid, spacer plates were machined and attached to the underside of the lid.
This ensured uniform contact between the fluid and the lid. A small clearance gap was left between the top of the
test article and the spacer plates. This ensured a free structural boundary condition at the top of the test article.
The entire experimental set-up (i.e., the test article, the acrylic tank, and the base) was assembled and placed on
two large foam blocks used to simulate free-free boundary conditions at the bottom of the steel base.

Before introducing any fluid to the system, a modal impact test was conducted on the test structure both with and
without the acrylic tank. Each measurement location required five averages with a frequency span of 0 to 6,400
Hz and 8,192 lines of resolution. The test was designed to characterize the responses of the test article and the
acrylic tank. The primary measurements were recorded along the top and center of both the test article and tank.
It was determined that the presence of the acrylic tank had little influence on the natural frequencies and modes
of the steel test article. Consequently, the analytical and finite element models developed here do not explicitly
model the acrylic tank.

Impact testing was only practical for the dry configurations. In the fluid-filled configurations, the test article was
excited with a stinger. The presence of the acrylic lid and the fluid made it impossible to attach an accelerometer
to the metal test article. Instead, a laser vibrometer was used to measure the response of the test article. Initially,
the scanning features of the vibrometer were used to scan roughly one third of the structure. This configuration
was adequate for determining low nodal diameter mode shapes, but modes with relatively high numbers of nodal
diameters could not be distinguished. To remedy this deficiency, the laser vibrometer was subsequently
configured in point-to-point mode. The vibrometer was physically moved or adjusted to measure single points 10°
apart along the top of the test article (a representative image of the vibrometer set-up can be seen in Figure 3).
For an approximately 30° sector of the test article, the stinger set-up blocked the path of the laser. It was
therefore not possible to measure points around the entire circumference of the test article. Nevertheless, the
available measurement points did allow for a reasonable determination of the mode shapes. The stinger itself
was not modeled in either the analytical or the finite element approaches. Thus, the mass loading effects of the
stinger are a potential source of error in the results. However, these effects were largely mitigated by placing the
stinger as close to the base of the cylindrical portion of the test article as possible. Evaporation of the alcohol was
observed to cause shifts in some of the resonant frequencies of the fluid structure system. Consequently, the
frequency response at each measurement point was compared to a baseline response. If significant frequency
shifts were observed at any point, the fluid cavities were topped-off with additional fluid and the response at the
given was measured a second time. This topping-off of the cavity with additional fluid was found to be an
effective means of removing any frequency shifts.

Figure 2: Experimental configuration with stinger in place

Figure 3: Laser vibrometer measuring response of steel test article

Three different fluid configurations were tested. In the first test, fluid filled the interior cylindrical cavity only. The
second test considered fluid in the annular cavity only. Fluid filled both cavities in the third test. Modal tests of the
three different fluid conditions were measured using a stationary-random input and a laser vibrometer that was
adjusted or moved to pre-determined measurement points. Each fluid condition required 50 averages at each
measurement point. The test article was excited with 0.5 lb rms random signal with frequency content between
500 and 6,400 Hz. The frequency response of the structure was processed with a Hanning window with 0%
overlap.

Figure 4: Experimental configuration with fluid filling entire cylindrical cavity

To identify mode shapes from the frequency response functions (FRFs), the impact measurements were fit with a
complex mode indicator function (CMIF) and the randomly excited responses were fit with both a CMIF and a
local least squares polynomial fit. The impact FRFs were very consistent between measurement locations. The
FRFs that were measured for the three different fluid configurations had some resonance shifting between FRFs.
In those instances, the local least squares method was used to identify the modal parameters.

V. Results and Discussion

Figures 5-7 are natural frequency maps for three different test configurations. Figure 5 represents the case in
which fluid filled the cylindrical portion of the test article only. In Figure 6, fluid is in the annular cavity only. Figure

7 considers fluid in both cavities. All frequency maps depict the lowest branch of uncoupled structural and
acoustic frequencies (shown in black). It can be observed that the experimentally determined structural natural
frequencies for the dry test article agree very well with those predicted by the in vacuo finite element model. The
in vacuo natural frequencies predicted by analytically modeling the test article as a clamped-free cylindrical shell
can be seen in Figure 5. For natural modes with circumferential wave numbers between one and four, it can be
observed that the in vacuo frequencies of the test article differ significantly from the analytical predictions for a
clamped-free shell. This is due to the fact that these modes are dominated by motion in the base of the test
article. Given this behavior, the analytical approach to predicting coupled frequencies is only expected to be valid
for coupled modes that have circumferential wave numbers greater than four.

In vacuo FE model of test article I

-B- In vacuo analytical model of clamped-free
cylindrical shell I

a Experimental test article in air I
H#— Rigid wall analytical acoustic I
0 Coupled fluid-structure analytical model I
a Experimental test article with fluid in clyindrical duct
-©- Coupled fluid-structure FE model I
In vacuo FE model of test article II
a Experimental test article in air 1 1
-©- Coupled fluid-structure FE model II
a Experimental test article with fluid in cylinder 1 1

Nodal Diameter

Figure 5: Frequency map of test article with and without fluid in the cylindrical cavity

The blue curves in Figures 5-7 depict the coupled frequencies as calculated using the FSI capabilities of ANSYS.
The green triangles in these figures represent the experimentally determined frequencies of the fluid-loaded
modes while the pink squares are the fluid-loaded frequencies that were calculated using the analytical approach
described in Section II.

In each of the fluid-loaded configurations, the lowest one nodal diameter mode is an acoustically dominated
coupled mode. When excited, such modes are characterized by a relatively small response on the part of the
structure. Consequently, it was not possible to confidently identify the coupled one nodal diameter modes from
the experimental FRFs of the structure.

Table 1 lists the percent differences between the analytically and experimentally determined frequencies.
Similarly, Table 2 lists the percent differences between the frequencies calculated with the finite element model
and those determined experimentally. Percent differences are tabulated for the dry configuration as well as the
three different fluid-loaded configurations. In the case of the analytically calculated frequencies (Table 1), the
percent differences associated with the fluid-loaded configurations are all lower than the corresponding
differences in the dry configuration. Furthermore, since the amount that the base of the test article participates in
a given mode decreases with increasing nodal diameter, the percent differences associated with the analytical
solution also decrease with nodal diameter. In the case of the finite element model (Table 2), the same trends are
not apparent. In general, the finite element model demonstrates smaller percent differences when calculating dry
frequencies than it does when calculating fluid-loaded frequencies. Secondly, the percent differences do not, in
general, decrease with nodal diameter. These two observations suggest an inherent uncertainty associated with
the unsymmetric eigensolver used to perform an FSI modal analysis within ANSYS. Thus, for the modes in which

the cylindrical shell idealization of the test article is valid, the analytical approach appears to be a more reliable
predictor of the fluid-loaded frequencies than does the FSI solution in ANSYS.

-©- In vacuo FE model of the test article I

▲ Experimental test aticle in air I

Rigid wall analytical acoustic I

□ Coupled fluid -structure analytical
model I

▲ Experimental test article with fluid in
annular cavity I

-©-Coupled fluid -structure FE model I

012345678

Nodal Diameter

Figure 6: Frequency map of test article with and without fluid in the annular cavity

Nodal Diameter

-©- In vacuo FE model of the test
article I

a Experimental test article in air

□ Coupled fluid-structure analytical
model I

-X- Rigid wall analytical acoustic for
cylinder I

Rigid wall analytical acoustic for
annulus I

a Experimental test article with
fluid in both cavities I

-©- Coupled fluid-structure FE
model I

Figure 7: Frequency map of test article with and without fluid in both cavities

Percer

it Differences

Nodal

Fluid in

Fluid in

Fluid in Both

Diameter

Dry

Cylinder Only

Annulus Only

Cavities

4

16 . 79 %

14 . 63 %

5 . 29 %

2 . 81 %

5

9 . 03 %

6 . 24 %

2 . 61 %

1 . 37 %

6

4 . 64 %

3 . 16 %

0 . 77 %

- 0 . 07 %

7

3 . 04 %

1 . 68 %

0 . 43 %

- 0 . 53 %

8

1 . 79 %

1 . 30 %

0 . 37 %

- 0 . 40 %

Table 1: Percent differences of analytically calculated frequencies

Percer

it Differences

Nodal

Fluid in

Fluid in

Fluid in Both

Diameter

Dry

Cylinder Only

Annulus Only

Cavities

2

0 . 54 %

3 . 10 %

- 12 . 42 %

- 9 . 31 %

3

- 0 . 12 %

0 . 84 %

- 18 . 77 %

- 15 . 22 %

4

0 . 56 %

- 0 . 25 %

- 7 . 77 %

- 12 . 05 %

5

1 . 04 %

- 0 . 97 %

- 3 . 60 %

- 8 . 07 %

6

1 . 78 %

1 . 01 %

0 . 72 %

- 4 . 96 %

7

1 . 79 %

2 . 91 %

5 . 63 %

- 2 . 02 %

8

2 . 26 %

5 . 22 %

3 . 14 %

0 . 99 %

Table 2: Percent differences of frequencies calculated using finite element FSI solution

VI. Conclusions and Future Work

For those modes that have circumferential wave numbers greater than four, the analytical method of calculating
coupled natural frequencies demonstrates excellent agreement with the experimental results. In fact, for the
circumferential wave numbers at which it is valid, the analytical approach showed better agreement with the
experimental results than the finite element model. The aforementioned wave number restriction was due to the
fact that the base of the test article participated heavily in the response of the modes with lower wave numbers.
The restriction is thus due to simplifications in the geometric modeling of the test article and should not be
interpreted as a manifestation of inadequacies in the mathematical formulation of the coupled acoustic-structure
problem.

Immediate future work includes the comparison of fluid-loaded experimental FRFs to those calculated using finite
element and analytical methods. Furthermore, analysis of the experimental FRFs has not conclusively revealed
the presence of acoustically-dominated coupled modes. Alternate methods of experimentally identifying the
frequencies associated with these modes are currently being explored. Other possible areas for future work
include a redesign of the test article to eliminate the aforementioned wave number restriction. Finally, the
relatively high natural frequencies of the test article and limitations on the excitation band of the shaker made it
difficult to obtain all but the very lowest branch of natural frequencies. Use of a shaker capable of exciting at
frequencies in excess of 6 kFIz may be of future interest.

VII. References

[1] R.B. Davis, Techniques to ^Assess Acoustic-Structure Interaction in Liquid Rocket Engines. Ph.D. thesis,
Duke University, Durham, NC, USA, 2008.

[2] E. H. Dowell, G. F. Gorman III, and D. A. Smith. Acoustoelasticity: General theory, acoustic natural
modes and forced response to sinusoidal excitation including comparisons with experiment. Journal of
Sound and Vibration, 52(4): 51 9-542, 1 977.

[3] R.B. Davis, L.N. Virgin, and A.M. Brown. Cylindrical shell submerged in bounded acoustic media: A
modal approach. AIAA Journal, 46(3):752-763, 2008.

[4] J. Callahan and H. Baruh. A closed-form solution procedure for circular cylindrical shell vibrations.
International Journal of Solids and Structures, 36:2973-3013, 1999.

[5] M. C. Junger and D. Feit. Sound, Structures and Their Interaction. Acoustical Society of America,
Melville, NY, 1993.

[6] D. T. Blackstock. Fundamentals of Physical Acoustics. Wiley, New York, NY 2000.